I fully admit that teasing out correlation and causality is very difficult in situations like this.
Let's say the first hour is 100% productive, the 2nd is 60%. Perhaps the third is 30% and the 4th 20%? But would they really undo the good of hours one and two?
The research quoted in a lot of the Deliberate Practice literature suggests we can only focus intensely for 4 hours a day. Then it's a question of wasted time, or harmful time.
So, I do think there is a limit for consciously doing hard math. Subconsciously, however, who knows what one's brain is doing? There's plenty of anecdotes about working hard on a problem for hours without apparent progress and then, suddenly, having the breakthrough insight during a walk or in bed, supposedly during a break of working on the problem. Famous anecdotes are Archimedes in bath and Kekule's dream about snakes and benzene.
Now, chances are these guys were still thinking of the problem (one advantage of theoretical work is that you can combine it with most low-effort activities) and nobody who gets such an epiphany knows whether just keeping churning would have led to the same result, possibly earlier, but I think that there is some truth in this. Just as running for 16 hours a day is not the best training for any race, it is good to have breaks from doing extensive math.
Also, in my experience, real understanding of mathematical concepts comes to me after I've studied it, when I'm thinking about the matter in the background while doing other things. It's plausible that, if I were to overextend myself, I'd lose focus and not assimilate that much later.
"elite athletes...perform better than the rest of us in yet another way. These athletes excel...in how fast their brains take in and respond to new information -- cognitive abilities that are important on and off the court." [1]
[1]: http://www.sciencedaily.com/releases/2013/03/130318151634.ht...
I've seen much less as research professors or computer programmers. This could also be because the time commitments required in school for these topics is inconsistent with what's required to do sports competitively.
I'm not saying that exercise is bad. I'm positive that it's good. It's just my perception that those who exercise more in their youth tended not to be the best students.
There's also the perception when you're younger (or at least distinctly for me and the people I grew up with), that being smart was 'boring' and being sporty was better, so they wouldn't apply themselves in lessons because it wasn't cool. I imagine a lot of them were plenty smart, but they might not have let on.
If the article actually compared the benefits of fitness to the benefits of playing music - now that could speak to your question.
What about playing an instrument? Don’t you have to use right and left brain for a stringed instrument?
Yes. That has clear cognitive functions that do crossover. Especially learning to play and read the music at the same time. But exercise is number one, diet number two and then social interaction. These are the important things for brain function.edit: maybe being good at both is a sign of an innate skill for abstraction, or maybe it's another way for a student to dive into a subject and grow new abstractions and reinforce his brain.
Now I think there may be some causality. Learning music is an intro to binary math. (2 half notes in a whole note. Two quarter notes in a half note.)
I haven't seen a good empirical study to try and split this apart.
Music is invisible at first, the journey between unconscious appreciation and the 'parsing' stage is long, and full of counter intuitive realizations, which to me, is the same whatever domain your try to understand.
Another parallel is the way we interface with these. There's the remote long round trip way and the direct tangible way. For computers : large systems requiring pauses in your knowledge acquisition, think ~minutes build times (this is the main view on computers, lisp OSes and smalltalk browsers are unknown to many) vs REPLs. In music there's music theory[1], lots of wasteful (borderline absurd) ceremony and delay before reaching to the music itself, and just following along, failing and trying again (here I think the most used one is the direct, you buy an instrument and "play" without real understanding, opposite of computers).
Hoping I wasn't too blurry.
[1] Have you seen Chris Ford Functional Composition talk ? https://www.google.com/search?q=chris+ford+functional+compos... (youtube/skillsmater hosted) He manage to layer music theory ideas in a very simple manner in one hour, with direct rendering of what they are. Much more efficient than what I could experience or see in music classes younger (I understand that kid psychology is different especially in groups). It's really not very profound and actually it won't teach you music, just reference ideas needed to then impregnate the whole subject through your won learning process (I believe it's a 10000hour thing).
In that regard, it would also be similar to the exercise comparison in that any single exercise is likely to have very limited impact, but a rounded exercise routine provides much stronger benefits.
Fallacy right there.